Is it obligatory to re-pray the Fajr prayer if the worshiper began two minutes before sunrise and the time expired during the recitation or bowing?

You are not required to repeat the Fajr prayer if the sun rises while you are praying, because your prayer is valid, sufficient, and fulfills the obligation, contrary to the Hanafi school of thought.

The correct view is that whoever catches one rak'ah of Fajr before the sun rises, their prayer is valid, based on the Prophet's (peace be upon him) saying: "Whoever catches one rak'ah of Fajr before the sun rises has caught Fajr."

It would have been advisable for you to shorten the prayer to catch a rak'ah before sunrise so that it would be performed on time, and to avoid the disagreement of those who say that the time is not caught unless a rak'ah is caught.

Your prayer is valid, and you are not required to repeat it, whether you caught a rak'ah or merely the opening takbir before the time expired.